Problem 31 - Algebra and Trigonometry

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x+2)^{8} $$

Step-by-Step Solution

Answer

The first three terms of the binomial expansion are \( x^8, 16x^7, 112x^6 \)

Step 1: Applying the Binomial Theorem

We know from the binomial theorem that \((x+2)^{8}\) expanded is \( \binom{8}{0} x^8*2^0 + \binom{8}{1} x^7*2^1 + \binom{8}{2} x^6*2^2 +... \). But we only need to find the first three terms.

Step 2: Calculating Binomial Coefficients

First, let's calculate binomial coefficients using combination formula. \( \binom{8}{0}=1 \), \( \binom{8}{1}=8 \) and \( \binom{8}{2}=28 \).

Step 3: Evaluating the first three terms

Substituting the binomial coefficients into the binomial theorem, our first three terms are:- \( \binom{8}{0}*x^8*2^0 = 1*x^8*1 = x^8 \)- \( \binom{8}{1}*x^7*2^1 = 8*x^7*2 = 16x^7 \)- \( \binom{8}{2}*x^6*2^2 = 28*x^6*4 = 112x^6 \).